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Continuity and Differentiability

Short Answer Type

551. If y=e^{a x}sin bx, prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight a dy over dx plus left parenthesis straight a squared plus straight b squared right parenthesis straight y equals 0

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552.

If space straight y equals straight x space log open parentheses fraction numerator straight x over denominator straight a plus straight b space straight x end fraction close parentheses comma space prove space that space straight x cubed fraction numerator straight d squared straight y over denominator dx squared end fraction equals open parentheses straight x dy over dx minus straight y squared close parentheses.

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553.

If space straight x equals tan open parentheses 1 over straight a log space straight y close parentheses comma space show space that space left parenthesis 1 plus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus left parenthesis 2 space straight x minus straight a right parenthesis dy over dx equals 0.

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554. If y=x^x, prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction minus 1 over straight y open parentheses dy over dx close parentheses squared minus straight y over straight x equals 0.

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555.

If space space space space straight y equals left parenthesis straight a plus straight b space straight t right parenthesis space straight e to the power of straight n space straight t end exponent comma space prove space that space fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight n dy over dx plus straight n squared straight y equals 0.

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556. $If space straight y equals fraction numerator straight a space straight x plus straight b over denominator straight c space straight x plus straight d end fraction comma space prove space that space 2 straight y subscript 1 space straight y subscript 3 equals 3 space straight y subscript 2 squared.$

Here space space straight y equals fraction numerator straight a space straight x plus straight b over denominator straight c space straight x plus straight d end fraction therefore space space space space space straight y subscript 1 equals fraction numerator left parenthesis straight c space straight x plus straight d right parenthesis. straight a minus left parenthesis straight a space straight x plus straight b right parenthesis. straight c over denominator left parenthesis straight c space straight x plus straight d right parenthesis squared end fraction equals fraction numerator straight a space straight d minus straight b space straight c over denominator left parenthesis straight c space straight x plus straight d right parenthesis squared end fraction therefore space space space space space straight y to the power of 1 equals left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space space space space space space space straight y subscript 2 equals negative 2 space straight c left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 3 end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis space space space space space space space space space straight y subscript 3 equals 6 space straight c squared left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 4 end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis straight L. straight H. straight S. equals 2 space straight y subscript 1 space straight y subscript blank cubed end subscript space space space space space space space space space equals 2 cross times left square bracket left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 2 end exponent right square bracket cross times left square bracket 6 space straight c squared left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 4 end exponent right square bracket space space space space space space space space space equals 12 space straight c squared left parenthesis straight a space straight d minus straight b space straight c right parenthesis squared left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 6 end exponent space space space space space space space space space equals 3 cross times left square bracket negative 2 space straight c left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 3 end exponent right square bracket squared equals 3 space straight y subscript 2 squared space space space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 2 right parenthesis right square bracket space space space space space space space space space equals straight R. straight H. straight S.

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557.

If space straight y equals log space left parenthesis 1 plus cos space straight x right parenthesis comma space prove space that space straight y subscript 1 space straight y subscript 2 plus straight y subscript 3 equals 0.

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558.

If space 2 space straight y equals straight x open parentheses 1 plus dy over dx close parentheses comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction space is space constant space.

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559. If y = (tan^-1 x)², then prove that (1 + x²)² y₂ + 2x(1 + x²)y_{1 = 2.}

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560.

If space straight y equals sin to the power of negative 1 end exponent straight x comma space then space show space that space open parentheses 1 minus straight x squared close parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x dy over dx equals 0.

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